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Mathematics. — “Multiple umbilics as singularities of the first 
order of exception on point-general surfaces’. Communicated 
by Prof. D. J. KorrpwrG and Mr. D. pr Lanes. 
(Communicated in the meeling of November 26, 1904). 
1. Let us suppose a point-general surface, i. e. general if considered 
as a geometrical locus of points, in whose Cartesian equation parameters 
appear; then for a continuous change of those parameters also the 
surface will in general vary continuously in shape). If then we fix 
our attention on any kind of singular points, plaitpoints, umbilies, 
etc. appearing on a point-general algebraic surface in finite number, 
it may happen during the deformation that two or more of those 
singular points coincide. Such a point where this takes place may 
be called a twofold or multiple singular point of that kind. 
Now such a coincidence may generally occur, as the results tell 
us, in more than one way. For some of these ways the coincidence 
depends on a single relation between the coefficients of the Cartesian 
equation being satisfied, whilst for others it depends on more suchlike 
relations. The former cases belong to the singularities of the first order 
of exception, the latter to those of a bigher order. It is only with 
the former that we shall occupy ourselves in this paper’). 
For plaitpoints the singularities of the first class, which must be 
regarded as multiple plaitpoints, were investigated by the first 
mentioned *). Two entirely different kinds of double plaitpoints were 
found (the homogeneous kind and the heterogeneous one) ; furthermore 
the points of osculation proved to be threefold plaitpoints, the nodes 
of the surface twentyfourfold plaitpoints. 
It seemed advisable to make an investigation also for other singular 
points. This we have done for the umbilics. The results obtained 
are communicated in this paper. For proofs and more elaborate 
considerations see the dissertation by the second mentioned Mr. D. 
DE LANGE issued recently. 
a. The double umbilic at finite distance. 
2. If we place the origin of a rectangular system of coordinates 
at an umbilie and if we use the tangent plane in this point as zy-plane 
1) See for more general considerations of the same kind as follow here: , Ueber 
Singularitäten verschiedener Ausnahmeordnung und ihre Zerlegung”, Math. Ann. 
41, p. 286—307 (1893). 
2) See for the reason why these are asking in the first place our attention the 
paper just quoted, on page 287. 
3) D. J. Korrewre, ,Ueber Faltenpunkte’, Wien. Ber. 98, p. 1154~1191, (1889) 
also Arch. Néerl. 24, p. 57—98, (1890). 
